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I rotate it on its axis at a constant angular velocity, and so the electric field

**E**produced is non-solenoidal and can be described as the negative gradient of some potential V(x,y,z).

The electric field is induced perpendicularly to the vector potential, as the vector potential points in the azimuthal (φ) direction and the electric field has no azimuthal (φ) components.

So some current is expected to develop at right angles to the vector potential.

However, when current develops at right angles to the vector potential of some source, that current does not induce net change of flux linkage directed on the magnet. Or does it?

Since the magnet is not one monolithic object, the subobjects (magnetic domains) which make it up have different velocities. In the frame of reference of one of these subobjects (magnetic domains), the external magnetic field

**B**from the plasma is transformed to a magnetic field

**B'**observed in its instantaneous co-moving inertial frame. This suggests that in the instantaneous co-moving inertial frame of a magnetic domain, the electric field

**E'**may have solenoidal components which are absent in the lab frame.

Since the curl of the magnetization

**M**gives us a magnetization current, analogous to an electric current, the emf induced on these "magnetization currents" through the closed-line integral of the electric field, corresponds to energy exchange through electromagnetic induction. Thus, while the lab frame could suggest that no EMF is occurring on these magnetization currents (i.e.

**curl E**= 0), in the local "material" frame of each magnetic domain, there is a

**curl E'**due to an apparent changing magnetic field.

Also, as each magnetic domain revolves around the collective axis, the transformation of the apparent

**E'**and

**B'**fields change with time. The time derivative of

**B'**would differ in the material frame from the lab

**B**field by the time derivative of -

**v x E**/ c^2 (in the non-relativistic approximation). I would assume that the changes of velocity

**v**of a magnetic domain with time, which leads to apparent changes in the externally applied magnetic field

**B'**(

**v**), does not generate a physical EMF on its magnetization currents, but that the apparent change of EMF due to changes in external field

**B'**(

**E**(t,

**r**(t))) observed in the instantaneous co-moving inertial frame (which is offset from the lab frame by instantaneous velocity

**v**), ultimately due to time and spatial variations of

**E**in the lab frame, would result in a physical EMF. If this is true, wouldn't that mean the time integral of the physical EMF on the magnetization currents could accumulate (volt-seconds in SI units) even though

**B**is constant (i.e. curl

**E**= 0)? To put this in another way, is it right to think that a changing magnetic field

**B'**would be observed for a looped magnetization current traveling at velocity

**v**perpendicular to an electric displacement current ∂

**E**/∂t, which would be subject to an EMF (closed loop integral of

**E'**(t,

**r**(t))) that is not apparent in the lab frame?

Would then the work done on the plasma through the potential V(x,y,z) produced by a cylindrically-symmetric magnet with constant angular velocity be possible by extracting energy from the magnetization currents of a magnetic domain through the volume integral of the time integral of

**curl M**dot

**E'**, disregarding those changes of

**E'**which are specifically due to time variation of the velocity

**v**of a magnetic domain?